arXiv subjects
Publications and source records attributed to Piotr Frackiewicz.
The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure nor in mixed strategies is given.
Games with unawareness model strategic situations in which players' perceptions about the game are limited. They take into account the fact that the players may be unaware of some of the strategies available to them or their opponents as well as the players may have a restricted view about the number of players participating in the game. The aim of the research is to introduce this notion into theory of quantum games. We shall focus on PQ Penny Flip game introduced by D. Meyer. We shall formalize the previous results and consider other cases of unawareness in the game.
The Li-Du-Massar quantum duopoly model is one of the generally accepted quantum game schemes. It has applications in a wide range of duopoly problems. Our purpose is to study Stackelberg's duopoly with incomplete information in the quantum domain. The result of Lo and Kiang has shown that the correlation of players' quantities caused by the quantum entanglement enhances the first-mover advantage in the game. Our work demonstrates that there is no first-mover advantage if the players' actions are maximally correlated. Furthermore, we proved that the second mover gains a higher equilibrium payoff that the first one.
The aim of this paper is to bring together the notions of quantum game and game isomorphism. The work is intended as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in strategic form.
The aim of the paper is to study the Bertrand duopoly example in the quantum domain. We use two ways to write the game in terms of quantum theory. The first one adapts the Li-Du-Massar scheme for the Cournot duopoly. The second one is a simplified model that exploits a two qubit entangled state. In both cases we focus on finding Nash equilibria in the resulting games.
The aim of this paper is to discuss in some detail the two different quantum schemes for duopoly problems. We investigate under what conditions one of the schemes is more reasonable that the other one. Using the Cournot's duopoly example we show that the current quantum schemes require a slight refinement so that they output the classical game in a particular case. Then we show how the amendment changes the way of studying the quantum games with respect to Nash equilibria. Finally, we define another scheme for the Cournot's duopoly in terms of quantum computation.
The Marinatto-Weber approach to quantum game is a straightforward way to apply the power of quantum mechanics to classical game theory. In the simplest case, the quantum scheme is that players manipulate their own qubits of a two-qubit state either with the identity operator or the Pauli operator $σ_{x}$. However, such a simplification of the scheme raises doubt as to whether it could really reflect a quantum game. In this paper we put forward examples which may constitute arguments against the present form of the Marinatto-Weber scheme. Next, we modify the scheme to eliminate the undesirable properties of the protocol by extending the players' strategy sets.
We present a quantum approach to a signaling game; a special kind of extensive games of incomplete information. Our model is based on quantum schemes for games in strategic form where players perform unitary operators on their own qubits of some fixed initial state and the payoff function is given by a measurement on the resulting final state. We show that the quantum game induced by our scheme coincides with a signaling game as a special case and outputs nonclassical results in general. As an example, we consider a quantum extension of the signaling game in which the chance move is a three-parameter unitary operator whereas the players' actions are equivalent to classical ones. In this case, we study the game in terms of Nash equilibria and refine the pure Nash equilibria adapting to the quantum game the notion of a weak perfect Bayesian equilibrium.
We generalize the concept of quantum Russian roulette introduced in [A.G.M. Schmidt, L. da Silva, Physica A 392 (2013) 400-410]. Our model coincides with the previous one in the case of the game with two players and gives the suitable quantum description for any finite number of players.
Iqbal and Toor [Phys. Rev. A {\bf 65}, 022306 (2002)] and [Commun. Theor. Phys. {\bf 42}, 335 (2004)] generalized the Marinatto-Weber quantum scheme for $2 \times 2$ games in order to study bimatrix games of $3 \times 3$ dimension, in particular the Rock-Paper-Scissors game. In our paper we show that Iqbal and Toor's generalization exhibits certain undesirable property that can considerably influence the game result. To support our argumentation, in the further part of the paper we construct the protocol corresponding to the MW concept for any finite bimatrix game that is free from the fault.
We present a scheme for playing quantum repeated 2x2 games based on the Marinatto and Weber's approach to quantum games. As a potential application, we study twice repeated Prisoner's Dilemma game. We show that results not available in classical game can be obtained when the game is played in the quantum way. Before we present our idea, we comment on the previous scheme of playing quantum repeated games.
The paper is devoted to quantization of extensive games with the use of both the Marinatto-Weber and the Eisert-Wilkens-Lewenstein concept of quantum game. We revise the current conception of quantum ultimatum game and we show why the proposal is unacceptable. To support our comment, we present the new idea of the quantum ultimatum game. Our scheme also makes a point of departure for a protocol to quantize extensive games.
We modify the concept of quantum strategic game to make it useful for extensive form games. We prove that our modification allows to consider the normal representation of any finite extensive game using the fundamental concepts of quantum information. The Selten's Horse game and the general form of two-stage extensive game with perfect information are studied to illustrate a potential application of our idea. In both examples we use Eisert-Wilkens-Lewenstein approach as well as Marinatto-Weber approach to quantization of games.
We present the unique solution to the Quantum Battle of the Sexes game. We show the best result which can be reached when the game is played according to Marinatto and Weber's scheme. The result which we put forward does not surrender the criticism of previous works on the same topic.
We demonstrate implementations of the Eisert-Wilkens-Lewenstein scheme be- yond normal-form games. The scope of our research includes decision problems, i.e., one-player extensive games. The research is based on the examination of their features when the decision problems are carried out via the EWL protocol. We prove that unitary operators can be adapted to play the role of strategies in deci- sion problems with imperfect recall. Furthermore, we prove that unitary operators provide the decision maker possibilities that are inaccessible for classical strategies.
We generalize a concept of classical finite extensive game to make it useful for application of quantum objects. The generalization extends a quantum realization scheme of static games to any finite extensive game. It represents an extension of any classical finite extensive games to the quantum domain. In addition our model is compatible with well-known quantum schemes of static games. The paper is summed up by two examples.
We study possible influence of not necessarily sincere arbiter on the course of classical and quantum 2x2 games and we show that this influence in the quantum case is much bigger than in the classical case. Extreme sensitivity of quantum games on initial states of quantum objects used as carriers of information in a game shows that a quantum game, contrary to a classical game, is not defined by a payoff matrix alone but also by an initial state of objects used to play a game. Therefore, two quantum games that have the same payoff matrices but begin with different initial states should be considered as different games.